Launching the client
The first step in any of this is of course launching the client. The WoL client runs on Java so if you don't have that installed then that is the first step. Clicking the Client link on the upper right corner of the site will download a ".jnlp" file that tells Java how to run the client.
If you can't see the Client link like above, make sure you are logged in, and your account is associated with a guild.

The download process is different between browsers, here are some screenshots from Chrome, Firefox, Opera, and Internet Explorer. If a browser allows you to open the file directly with Java, that is usually easiest.Chrome:
Chrome will display a security warning
After clicking save simply click on the file to launch it

Firefox:
Firefox allows you to either directly open the file with Java, or save it to disk.

Opera:
Opera also allows you to either directly open the file with Java, or save it to disk.

Internet Explorer:
Internet Explorer's behavior seems to change with every version. For the currently released version 8, the dialog looks like this:
For version 9 a bar will appear at the bottom of your screen that looks like this:

If you chose to save the file, simply double click on the file from wherever you saved it.
If you don't have Java installed your browser will not offer you the option of opening the file with it, and manually clicking on the file will give you something like this:

Set up
Once you have the client running, you need to do a tiny bit of set up. Click on Edit -> Preferences. You should be looking at a screen that looks like this:
The top part allows the client to login as you and upload reports to your guild. Fill in your WoL account information here and click "Test account settings". This will do a test login to verify that the credentials you entered are correct. When a valid login has been entered the information is saved and doesn't need to be changed unless you want to upload to a different account.
The lower part tells the WoL client where your WoW log directory is. If this isn't filled in press the "Find/Test and save" button, locate and select the folder. Once this has been entered you shouldn't need to change it.

Now that things are all set up and running, let's move on to how to actually use the client.

Uploading a Log
The WoL Client provides 3 different options for uploading a log.

Open a file: This option brings up a file selection dialog that allows you to select an arbitrary file for parsing and uploading. This is useful if you need to re-upload a previously submitted log for any reason, among other uses.

Open the WoW Log: This option automatically selects WoWCombatLog.txt in the configured log directory.

Start a Live Report Session: This will start watching the WoWCombatLog.txt file and upload changes to the server as it sees them.

Note how at the bottom of the window it shows the currently logged in user. This provides a nice check as to whether you are properly logged in, and to what account the client currently is set up. If you want to upload to multiple guilds simply change the login in the client before uploading a report. This way you can have a personal guild for pugs or whatever you want.

Troubleshooting
So you have read through all the steps above... but things still aren't working quite right? Here are a few common problems that people have.

Client launching issues
There are a few different reasons for this that people ask about all the time. The most common is that people haven't installed Java. So if you haven't installed it... do so now! If you aren't sure, install it anyway as this will make sure you have the latest version.

Another issue that people have occasionally is the .jnlp file isn't associated with anything, even though they have Java installed. To fix this hold Shift and right click on the file -> Select "Open with...". You now need to browse to where Java is installed on your computer. There are two different versions of Java, 64 bit and 32 bit. Here are the default locations for these versions:
32 bit windows:

32 bit (x86): C:\Program Files\Java\jre6\bin\javaws.exe

64 bit windows:

64 bit (x64): C:\Program Files\Java\jre6\bin\javaws.exe

32 bit (x86): C:\Program Files (x86)\Java\jre6\bin\javaws.exe

If you are unsure what type of windows you have you can find out by hitting the start menu -> Right clicking on (My) Computer -> Properties. You should see something like this:

If these don't get the client running, your local cache may possibly be corrupted. Open the control panel -> Java (you may need to search for it) -> Temporary Internet Files -> Settings -> Delete Files.

Log uploading problems
One of the most common things seen is logs being uploaded to the "wrong" guild. If this is happening to you please check the login of the client as it doesn't follow the website. Check your login credentials as shown at the end of the Launching the Client post.

Some common things wrong with the actual log is it being too long in time. If you are seeing an error like the following then this is what is going on:
or alternatively like this one
These both mean that the log file you are trying to upload has entries from too long a time period. Usually this happens when you don't clean up the log file between raids. If you run into this you need to split the file up into separate files for each raid. The WoL client has a tool built in that does just this

In the WoL client click on Tools -> Split & zip logs. Select your WoWCombatLog.txt. The program will split the log up whenever it finds a gap in time of inactivity. I believe the gap must be at least 20 minutes with no log lines. A folder called "archive" will be created in the same directory as the log file, inside this folder will be zip files named YYMMDD HHMMSS.zip, with the time stamp based on the first logged line in there. To use this to upload after splitting it up simply extract the zip files in the log directory and tell the client to upload the .txt files that are created. Each file should be usable in the client.

So you have read all the common issues above... but nothing sounds like your problem? Turn on the Java console which will output some useful debugging information, and make a post in the help or bug reports forum

Other client issues
If you run into something not already described and the error doesn't make sense (or there is no error) then we are going to need more information. The first step is to turn on the Java console if it isn't already turned on. To do this go to the control panel -> Java -> Advanced tab -> Show console. This will cause the Java console to show up whenever you launch the WoL client. If you can't find Java in the control panel, try searching for it in the box on the upper left. The window looks like this:
Debug text and any errors will be printed to this screen as the client does it's work. After enabling the console try to reproduce your error and copy the text from the console to a post on the forums.
A copy of the raw log file you are working with might also be helpful. If you can try to keep it around until the problem is fixed.

Miscellaneous
One thing that might help if you use the client a lot is being able to "install" it. Once you have ran the client at least once you can do this by opening the Java control panel -> General tab -> View... in the Temporary Internet Files section -> Select World of Logs Client -> Install shortcuts. This will put a shortcut to the client on your desktop and the client will be kept up to date for you.